Ten hours before their band are announced as the joint most-nominated act for the 2013 Grammy Awards, GQ’s sixth-worst dressed man of 2010 and Movember’s 12th-highest UK fund-raiser are together recalling their audience with the leaders of the free world.

“We rock up at the back door and all our names are wrong,” says Winston Marshall, Mumford & Sons’ habitually scruffy 24-year-old banjo player. He’s describing the British folk-rock outfit’s less than auspicious arrival at a state dinner in Washington, held in March by President Barack Obama in honour of the official visit of Prime Minister David Cameron. Mumford & Sons and the American soul singer John Legend were the musical turns; the guests were the great and good of the UK entertainment world, including Hugh Bonneville, Idris Elba and Damian Lewis.

“So security have to recheck everyone,” continues Marshall. “We’re there for half an hour. All our birthdays were out by one day. And then Idris Elba walks past, in a tux, no tie, Ray-Ban Wayfarers on, straight through – they didn’t even check him!” he laughs of the Hackney-born actor. “And the security were like, ‘yes sir!’ We couldn’t believe it. They were completely in awe of him.” As were the band, in a way – the foursome would later ask Elba to star in and direct the video for their recent single Lover of the Light; the deal was concluded over pints in a pub near the Shepherd’s Bush home that singer Marcus Mumford, 25, shares with his wife, the actress Carey Mulligan.

Truth be told, the security men should have had no trouble recognising them. Mumford & Sons have sold four million copies of their 2009 debut, the Brit Award-winning Sigh No More. In America, this year’s follow-up, Babel, sold a staggering 600,000 copies in its first week of release, almost double what Justin Bieber achieved with his second album; only Taylor Swift bested them this year. In little more than two months, Babel has already sold two million copies, making Mumford & Sons just about the biggest British band in the world right now. None the less, playing in the White House must have been a surreal experience.

“It was,” nods keyboard player Ben Lovett, 26, twirling the moustache that helped him raise £7,000 in last month’s cancer awareness facial-hair charity drive. “Although I was more freaked out for Damian Lewis! He was sitting with Michelle Obama.” The Homeland star recently told me that Legend was invited to sing by President Obama. But that Mumford & Sons’ performance came courtesy of a personal request from Samantha Cameron.

To a man, Mumford & Sons squirm when I tell them this. They’re possibly still smarting from the revelation, made in The Sunday Telegraph, that the band are the Prime Minister’s favourite band. Even if they don’t care about being cool – and the waistcoat-wearing, hoedown-loving Mumfords are stoutly and admirably uninterested in such notions – being tarred as the Conservative leader’s fave rave, and that of his wife, seems beyond the pale.

“That’s not how we heard it,” says Lovett.

“Yeah, there are lots of different versions of events,” adds bass player Ted Dwane, 28.

“Um,” demurs Mumford, “that was never made clear to us.”

“We were invited by the White House,” insists Marshall.

“By the President,” nods Mumford.

I raise what I hope is a querulous eyebrow. Lewis was quite insistent his intelligence was correct. And he is the Emmy-winning lead actor in President Obama’s favourite television show. And he was sitting at the top table.

Plus, he and David Cameron are both Old Etonians. If anyone had insight into Samantha Cameron’s musical taste and influence, it would surely be him.

“She said to me, ‘have you heard First Aid Kit?’” continues Mumford. First Aid Kit are two young country-folk Swedish sisters, whose second album The Lion’s Roar is one of the year’s best – if more obscure – releases. “I was like, ‘what the ----!’ She said that to me!” Just so we’re clear: the Prime Minister’s wife asked Mumford if he’d heard of First Aid Kit? Even if Mumford & Sons weren’t privately schooled and quite posh themselves, that’s surprising.

“Yes! I know! Exactly! That’s why I was so baffled,” smiles the supremely affable Mumford, an alumnus (like Lovett) of Wimbledon’s King’s College.

“But it’s exactly what she said. She went way up in my estimation, from, well, yeah,” he mutters, indicating a low base with his outstretched arm.

Anyway, whatever, shrugs Lovett. They were thrilled that their reputation as one of the best live bands around had reached high up the political totem pole, on whichever side of the Atlantic – and on whatever wing of the political spectrum – it happened to be.

It’s a rainy, bitter Thursday in Glasgow. The four members of Mumford & Sons are sprawled in coats and scarves in the backstage catering area of the cavernous Scottish Exhibition and Conference Centre. Remarkably, they can remember every single appearance they’ve made in the city since forming in London in 2007.

It’s a mark of their dedication to the life of the touring rock band that they can both name and sequence all the venues: Captain’s Rest (“30 or 40 people in the bottom of a pub”), Nice ‘N’ Sleazy, Òran Mór twice, King Tut’s Wah Wah Hut, the ABC, the O2 Academy. And, neatly capping off this steady ascent up the rungs of the UK concert circuit, the night before we meet Mumford & Sons are announced as the first headline act at next summer’s T in the Park, Scotland’s national music festival.

“I think that’s quite a good indication of why things to us haven’t felt really rushed or really forced upon us,” considers Mumford of the band’s progress. Their success is not about hipness or hype (they’re too folky for that), or hits (the “big” first album single The Cave only reached Number 31), or pop star looks (although these well-spoken Londoners have a cheery, grizzled charm).

Their achievements in practically every corner of the world have come the old-fashioned way: word of mouth and performing. People might question these middle-class English urbanites’ folk and country music credentials. But there’s no denying the force, vigour, conviction, passion and exuberance of their songwriting, four-part harmonies and onstage hootenannies.

“We’ve been touring the whole time,” continues the singer. (The band carry his name not because he’s the boss, but because they liked how Mumford & Sons was suggestive of a trustworthy, old-fashioned family business.) Some people are like, ‘wow, it’s gone so quickly’ – and it has. But in most places it’s felt progressive. It’s gone from the dingy little pub to the sort-of club to the theatre. We haven’t skipped a rung.” Indeed, Marshall is so in love with life on the road that he has the word “Tour” tattooed on his left arm.

It’s now late afternoon on the second date of the second leg of Mumford & Sons’ Tour of Two Halves UK tour. The opening instalment of the round-Britain peregrination took in theatres. This new run of shows sees the band performing in the country’s biggest, draughtiest arenas. In an ideal world, these barn dance-friendly, pub-loving musos would always perform in theatres.

So playing a concrete shed like the 10,000-capacity SECC and bringing in elaborate lights and screens is resolutely not, says Lovett, about “the wow factor – check us out, we’re doing really well now”. It is, he notes, about the opposite.

“It’s more like, ‘let’s make this room feel half the size it is. Let’s really include the people who are 50 metres away from us’. That’s the challenge a lot of bands face when they’re trying to …” he tails off. Unlike the reach-for-the-stars likes of Coldplay and U2, but rather like the more questioning Radiohead and Arcade Fire, issues of size and status always give Mumford & Sons pause. Instead of doing the SECC, “our other option would be to do three or four O2 Academy-size shows. So it’s a balance of ------- people off and having inflated ticket prices and really small gigs, or coming and trying to play the right room the right way.”

The band recently posted a blog on their website about ticketing. A batch of fake tickets were sold to fans for their Portsmouth show. They’re determined to avoid that happening again. The canny, forward-thinking Mumford & Sons made Sigh No More themselves before licensing it to a record label, and apply a gimlet eye to how they’re marketed, so they’re not about to let this go. They’re trying to take control of all their ticket sales.

In a time when the Rolling Stones charged a basic £106 per ticket for their recent reunion shows, is it important for the band to manage prices? “Yeah,” replies Mumford, “we have a discussion before every tour about ticket prices. We always want to be just below average.” It’s been said before. “Yeah, that’s how we approach albums!” he laughs. Yet for all his readiness to self-deprecate, Mumford is careful to avoid upsetting the horses. When asked if he thinks the Stones’ ticket prices were justified, he shifts uncomfortably. “I don’t know,” he says, three times. “Of course it’s really easy to say it sounds really greedy. It’s not our place to comment. It’s the ------- Stones, innit?” He pauses. “Makes you raise your eyebrows though, doesn’t it?”

Mumford is a thoughtful, bookish chap from a learned background. He briefly read Classics at Edinburgh University, before abandoning his degree to concentrate on music. His brother, James, has a PhD from Oxford and is a senior researcher with the independent Westminster think tank The Centre for Social Justice. His parents were founders of the Vineyard Church in Britain, and Mumford was born in California, base of the organisation, where his parents were working “for a couple of years”. It’s a “neocharismatic evangelical Christian denomination”. He denies that there are too many religious elements to his lyrics, despite the Biblical title of the second album.

Still, given the good works of his family, does he try and bring that ethos to bear on how Mumford & Sons conduct themselves? “I’m certain that it influences me, but I don’t set out to try and be like them in that sense. But I’m certain that it’s influenced my character, as anyone who loves their family would be. But that’s just who we are as a band. All four of us want to treat our crew fairly and treat other bands fairly.” And how does he feel about women bishops? Mumford initially bursts out laughing at the question. (It’s not a total non-sequitur: the incoming Archbishop of Canterbury is a one-time worshipper at Holy Trinity Brompton, a church associated with the Mumford family.) But then he answers: “I think they should have them, definitely. And the way I understood it from the press I read was that they’re not voting against having them – they were voting against a clause in the motion that controls how women priests in that diocese are allowed to vote on who becomes bishop. They were voting against a technicality. But of course it’ll take them 100 years to vote again.”

Meanwhile, on the other establishment topic du jour, Mumford is forthright. “The Royal pregnancy? I’m sorry, I’m sorry – but it’s ------- awesome.” Why? Is it the good news cheer that an embattled country needs? “I probably have different views on the monarchy than everyone else. A lot of people are very surprised sometimes [at my view]. I looked at a picture of Winston Churchill in 1945 opening a door for the Queen, and welled up.” Then he’s pro-monarchy? “I love the Royals. I think they work ------- hard. And I think they do really good things for our country.”

One thing Mumford won’t discuss is his wife. He and Mulligan seemingly knew each other, through church circles, in childhood. They began dating in summer 2011 and were married in Somerset last spring. I ask his band mates: in this none-more-down-home group, now that their friend and frontman is married to a “Hollywood A-lister”, do they have to work harder to protect his privacy?

Lovett: “Yeah. I mean, we conduct ourselves like we always have. We’re very specific about who we talk to, about anything. We don’t do [Never Mind the] Buzzcocks. We try and keep a vaguely low profile. We’re not pop stars. So that stuff is not really a big deal. People don’t camp outside their [Mumford and Mulligan’s] house. Although I know they had a photo in Hello! or something – my mum said, ‘oh, I saw Marcus and Carey’. But there was no front page ‘couple of the summer!’ or stuff like that.”

Dwane: “Yeah, kinda. I think Marcus is pretty good at keeping all that sort of stuff in check. We all help each other out in that stuff. But I think a lot of the time ----’s happening that we don’t even know about. We’re not on the internet checking what’s happening with Mumford & Sons around the world – we’re on the internet checking out what’s happening with the results to see how our fantasy football’s doing. We’re in the eye of the storm – people might be frenzied around us, but luckily we don’t have to deal with it really. The paparazzi doesn’t happen that much.”

Marshall: “Um, I probably want to avoid those questions too. Not to be too secretive about it, but it is what it is.”

Later that evening, the news comes in from Los Angeles: Mumford & Sons have been nominated in six categories at the 55th Grammys, which are awarded in February, including Album of the Year and (oddly, perhaps; or maybe not that oddly) Best Americana Album.

The band won’t immediately notice – they’re too busy bringing good cheer to the SECC, inducing a mass céilidh singalong. Nor, one suspects, will they care too much.

Earlier, I had asked each of them about their ambitions for the band. “Just keep doing it,” replied Mumford. “Honestly. Do more shows. Record more albums. Write more songs. I feel like we’ve got a lot more to say. We haven’t said a third of it, a half of it, an eighth of it yet.”

“Hopefully,” muses Lovett, “one day people will be able to look at Mumford & Sons and say, ‘that’s a career band’. It’s all about time instead of sales.”

“It’s something we’re working out,” admitted Dwane. “’Cause we had really humble aspirations. I remember our first EP launch at [London venue] The Luminaire. That was mind-blowing – 250 people in your own town!”

For Marshall, the answer is simple: “Tour.”

Mumford and Sons play the O2 in London tonight. A special edition of 'Babel', and a live DVD, 'The Road to Red Rocks', are both available now. See mumfordandsons.com